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1.
We derive a representation of orthogonal matrix polynomials on the real line as product of certain Schur complements and obtain matricial generalizations of well-known formulas for orthogonal scalar polynomials associated to a polynomial modified moment sequence. 相似文献
2.
3.
E. Bourreau 《Acta Appl Math》2000,61(1-3):53-64
In the scalar case, computation of recurrence coefficients of polynomials orthogonal with respect to a nonnegative measure is done via the modified Chebyshev algorithm. Using the concept of matrix biorthogonality, we extend this algorithm to the vector case. 相似文献
4.
We study two infinite sequences of polynomials related to Jordan blocks that have various interesting properties. We show that they are orthogonal polynomials whose sequences of moments are Catalan numbers and we relate them explicitly to the Chebyshev polynomials. We also use them to compute the singular values of some Jordan blocks. Finally, we investigate some combinatorial properties of the inverse sequences of these polynomials; we show them to be intimately related to the convolutions of the Catalan sequence. 相似文献
5.
In this article, we investigate orthogonal polynomials associated with complex Hermitean matrix ensembles using the combination of the methods of Coulomb fluid (or potential theory), chain sequences, and Birkhoff–Trjitzinsky theory. We give a general formula for the largest eigenvalue of the N×N Jacobi matrices (which is equivalent to estimating the largest zero of a sequence of orthogonal polynomials) and the two-level correlation function for the α ensembles (α>0) introduced previously for α>1. In the case of 0<α<1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems. 相似文献
6.
Antonio J. Duran 《Journal of Approximation Theory》1999,100(2):2239
Ratio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind. 相似文献
7.
P. López-Rodríguez 《Constructive Approximation》1999,15(1):135-151
We describe the image through the Stieltjes transform of the set of solutions V of a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of
V for which the matrix polynomials are dense in the corresponding
2
space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set.
May 20, 1997. Date revised: January 8, 1998. 相似文献
8.
In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu
and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal
polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic
spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an
integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results
establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which
have not been explored so far in the literature. 相似文献
9.
We give conditions for the coefficients in three term recurrence relations implying nonnegative linearization for polynomials
orthogonal with respect to measures supported on convergent sequences of points. The previous methods were unable to cover
this case.
January 14, 1999. Date revised: January 31, 2000. Date accepted: October 24, 2000. 相似文献
10.
Hua Dai 《计算数学(英文版)》2004,22(5):671-680
Let $P$ be an $n\times n$ symmetric orthogonal matrix. A real $n\times n$ matrix $A$ is called P-symmetric nonnegative definite if $A$ is symmetric nonnegative definite and $(PA)^T=PA$. This paper is concerned with a kind of inverse problem for P-symmetric nonnegative definite matrices: Given a real $n\times n$ matrix $\widetilde{A}$, real $n\times m$ matrices $X$ and $B$, find an $n\times n$ P-symmetric nonnegative definite matrix $A$ minimizing $||A-\widetilde{A}||_F$ subject to $AX =B$. Necessary and sufficient conditions are presented for the solvability of the problem. The expression of the solution to the problem is given. These results are applied to solve an inverse eigenvalue problem for P-symmetric nonnegative definite matrices. 相似文献
11.
A Necessary and Sufficient Condition for Nonnegative Product Linearization of Orthogonal Polynomials
Ryszard Szwarc 《Constructive Approximation》2003,19(4):565-573
A necessary and sufficient condition for nonnegative product linearization of an orthogonal polynomial system is derived. The method goes through a discrete hyperbolic boundary value problem associated with the three-term recurrence relation. The paper provides a unified approach to nonnegative linearization problems that comprises results obtained earlier and gives new ones. 相似文献
12.
Compatibility of a Hankel n × n matrix H and a polynomial f of degree m, m ? n, is defined. If m = n, compatibility means that HCTf=CfH where Cf is t companion matrix of f. With a suitable generalization of Cf, this theorem is generalized to the case that m < n. 相似文献
13.
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrixrnew rows and columns, so that the original Jacobi matrix is shifted downward. Thernew rows and columns contain 2rnew parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials, and the 2rnew parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the 1?orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials. 相似文献
14.
The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order
differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix
W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of
some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical
functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer
parameters to obtain an immediate extension beyond these cases. 相似文献
15.
Rostyslav Kozhan 《Constructive Approximation》2012,36(2):267-309
We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L 1-type condition on Jacobi coefficients, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik and Simon (Int. Math. Res. Not. 32:19396, 2006). The above results allow us to fully characterize the Weyl?CTitchmarsh m-functions of Jacobi matrices with exponentially converging parameters. 相似文献
16.
In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not. 相似文献
17.
We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size 2 × 2 satisfying second-order
differential equations with polynomial coefficients. We consider here three one-parametric families of weight matrices, namely,
and
and their corresponding orthogonal polynomials. We also show that the orthogonal polynomials with respect to the second family
are eigenfunctions of two linearly independent second-order differential operators. 相似文献
18.
《复变函数与椭圆型方程》2012,57(9):745-759
In this paper we will discuss the problem of generation of sequences of orthogonal polynomials with respect to measures supported on the unit circle from a given sequence of orthogonal polynomials using a perturbation of a cubic sieved process. The basic tools are the Szeg? forward recurrence relation as well as the fact of the coprimality of orthogonal polynomials on the unit circle and their corresponding reverse polynomials. We also give the connection between the associated orthogonality measures. Finally, some examples of this cubic decomposition are shown. 相似文献
19.
We develop structural formulas
satisfied by some families of
orthogonal matrix polynomials of size $2\times 2$ satisfying
second-order differential equations with polynomial coefficients. We consider
here two one-parametric families of weight matrices,
namely
\[
H_{a,1}(t)\;=\;e^{-t^2} \left( \begin{array}{@{}cc@{}}
1+\vert a\vert ^2t^2 & at \\bar at & 1 \end{array} \right) \quad {\rm and} \quad H_{a,2}(t)\;=\;e^{-t^2} \left( \begin{array} {@{}cc@{}}
1+\vert a\vert ^2t^4 & at^2 \\bar at^2 & 1
\end{array} \right),
\]
$a\in \mbox{\bf C} $ and $t\in \mbox{\bf R} $, and their corresponding orthogonal
polynomials. 相似文献
20.
An alternative to Lagrange inversion for solving analytic systems is our technique of dual vector fields. We implement this
approach using matrix multiplication that provides a fast algorithm for computing the coefficients of the inverse function.
Examples include calculating the critical points of the sinc function. Maple procedures are included which can be directly
translated for doing numerical computations in Java or C.
A preliminary version of this paper has been presented at AISC 2006. 相似文献